We could derive from Corollary REF  and [1]} a lower bound of the form \(n^{3/4+o(1)}\)  on the mixing time in total variation for the lazy random walk. However, \(3/4\)  is in all likelihood not the optimal exponent: [2]} proved that the simple random walk on the uipt (the local limit of uniform triangulations as their size goes to infinity) travels a distance \(t^{1/4+o(1)}\)  after time \(t\) . We thus expect that the mixing time should be of order at least \(n^{1+o(1)}\) . Indeed, roughly speaking, since a map with \(n\)  faces has diameter \(n^{1/4+o(1)}\) , we need to wait for a time \(n^{1+o(1)}\)  before the random walk has a chance to explore the whole map.
